Assume that \([\xi ]\in \mathsf {Ext}^{1}_{R}(X,T^{\Lambda })\)  where \(\xi :0\rightarrow T^{\Lambda }\rightarrow N\rightarrow X\rightarrow 0\) . Let \(\lambda \in \Lambda \)  and \(\pi _{\lambda }:T^{\Lambda }\rightarrow T_{\lambda }\)  be the \(\lambda \) -th projection. Consider the push-out
\({\xi :& 0[r]&T^{\Lambda }[d]_{\pi _{\lambda }}[r]&N [d][r]&X@{=}[d][r]&0\\(\pi _{\lambda })_{\sharp }(\xi ):&0[r]&T_{\lambda }[r]&U [r]&X[r]&0 }\) 
\((\pi _{\lambda })_{\sharp }(\xi )\)  denotes by the bottom row exact sequence of the above diagram.
It is well-known that the group homomorphism \(\Phi :\mathsf {Ext}^{1}_{R}(X,T^{\Lambda })\rightarrow \mathsf {Ext}^{1}_{R}(X,T)^{\Lambda }\)  given by \(\Phi ([\xi ])=\lbrace [(\pi _{\lambda })_{\sharp }(\xi )]\rbrace _{\lambda \in \Lambda }\) , where \([\xi ]\in \mathsf {Ext}^{1}_{R}(X,T^{\Lambda })\) , is an isomorphism. If \([\xi ]\in \mathsf {Ext}^{1}_{\sigma [M]}(X,T^{\Lambda })\) , then \((\pi _{\lambda })_{\sharp }(\xi )\in \mathsf {Ext}^{1}_{\sigma [M]}(X,T)\)  by [1]}. Hence, we get the restriction of \(\Phi \)  on \(\mathsf {Ext}^{1}_{\sigma [M]}(X,T^{\Lambda })\) , denoted by \(\Psi :\mathsf {Ext}^{1}_{\sigma [M]}(X,T^{\Lambda })\rightarrow \mathsf {Ext}^{1}_{\sigma [M]}(X,T)^{\Lambda }\) . Then there is a commutative diagram
\({0[r]&\mathsf {Ext}^{1}_{\sigma [M]}(X,T^{\Lambda })[r][d]^{\Psi } & \mathsf {Ext}^{1}_{R}(X,T^{\Lambda })[d]^{\Phi }\\0[r]&\mathsf {Ext}^{1}_{\sigma [M]}(X,T)^{\Lambda }[r] & \mathsf {Ext}^{1}_{R}(X,T)^{\Lambda } }\) 
where all rows are exact. Therefore, we know that \(\Psi \)  is injective.
